Exact Hessian Cancellation and Quartic Nondegeneracy at the Round Circle

We study the decomposed Möbius energies $E_1$ and $E_2$ at the round circle and determine their complete normal Hessians: $H_1(C)=2\pi (\mathrm{\mid }{D}_s{\mathrm{\mid }}^3-\mathrm{\mid }{D}_s\mathrm{\mid })I_2$ and $H_2(C)=-(4\pi \mathrm{/}3)(\mathrm{\mid }{D}_s{\mathrm{\mid }}^3-\mathrm{\mid }{D}_s\mathrm{\mid })I_2$. Thus $H_1(C)=3H_E(C)$ and $H_2(C)=-2H_E(C)$, where $H_E(C)$ is the Hessian of the full Möbius energy. This exact proportionality singles out the Möbius-invariant combination $F=2E_1+3E_2+2{\pi }^2$. We prove that $F(C)=0$ and that its first three derivatives vanish on the full space of parametrized variations. For scalar binormal variations, the fourth variation has a Fourier tensor with complete $3+1$ resonance cancellation and finite $2+2$ fixed-sum blocks. Their positivity follows from an explicit weighted-difference tridiagonalization and a closed determinant formula. For arbitrary normal variations in ${\mathbb{R}}^3$, Möbius invariance in ${\mathbb{R}}^4$ reduces the radial component to a second binormal direction. The resulting two-component quartic form splits into trace, symmetric-traceless, and antisymmetric blocks, which are positive on the active coordinates. Consequently, $D^{4}F(C)[u,u,u,u]\ge 0$ for every smooth real normal field $u$, with equality exactly on the normal tangent space of the Möbius family of round circles. This yields a fourth-order nondegeneracy phenomenon that is not determined by the total-energy Hessian alone.